Consider the following PDE in 3 dimensions $$ \epsilon^{ijk}\partial_j B_{kl}(x) = \delta^i_l\delta^{(3)}(x)$$ Does $B_{kl}(x)$ have a solution? (It can have any kind of singularity, e.g. it can have singularity at the positive $z$ axis like a Dirac string emanating from the origin.)

  • $\begingroup$ Is $B_{kl}$ antisymmetric? $\endgroup$ – Qmechanic Nov 15 '18 at 9:01
  • $\begingroup$ There is no such constraints $\endgroup$ – Weicheng Ye Nov 15 '18 at 12:09
  • $\begingroup$ Then the eq. system decouples in 3 independent subsystems: $l=1$, $l=2$, & $l=3$. $\endgroup$ – Qmechanic Nov 16 '18 at 12:26

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